It seems that, since mourning is prohibited during the month of Nisan, one could not observe the restrictions of sefirah until the end of Rosh Chodesh Iyar, since one is also not permitted to mourn on Rosh Chodesh. Yet, we can see that many communities have the kosher custom to observe the mourning of sefirah during the second half of Nissan, even ones that do not say tachnun at any point during the month! How can this be?
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There are various "degrees" of "mourning". Generally, the most noticeable change is refraining from hearing music, which, BTW, is not a minhag mentioned in the Shulchan Aruch. It developed years later and has become a common minhag. (After the destruction of the Bet Hamikdash, one should NEVER listen to music. Few follow this halacha.) Generally, the strict laws of mourning are similar to those done during the 9 days such as not eating meat, not drinking wine, etc. I'm not sure that refraining from music would be considered a form of "mourning" in the strict technical sense. Tachanun, BTW, is not said during both happy occasions as well as sad occasions - it's not said in a mourner's house. So Tachanun cannot be used as your example of implying that it's ommitted only because of happy occasions.